Given that where and , prove that:
a) if u varies but v is constant, then the locus of is a parabola,
b) if v varies and u is constant, then the locus of P is again a parabola.
Show that these two parabolas have the same axis and focus.

From the relation:
Which is the parametric equation for a parabola.
When u varies, the parabola has focus and axis .
When v varies, the parabola has focus and axis .

The parabolas face in opposite directions, and both cross the y-axis, assuming that the constants are greater than 0. My problem is how do I show that they have the same focus. They do have the same axis.
Thanks!

Sep 30th 2010, 05:21 AM

HallsofIvy

Just to make it clear that "v is a constant", write c in place of v.

and .

From the second equation,

Put that into the first equation: which is the equation of a parabola with vertex at , axis along the x-axis, and opening to the right. You should have learned that the parabola has focus at (0, c). Adapt that to this problem.